abstract algebra equivalence relations

Question

Let R be the set of real numbers. For a,b∈R define a∼b if a − b ∈ Z. Prove that ∼ is an equivalence relation.

I know you have to prove that it's reflexive, symmetric, and transitive I just don't know how to show my work.

Answer 1

Let $x\sim y$ thus $x-y\in\mathbb Z$, which gives $y-x\in\mathbb Z$, which says $y\sim x$.

$x-x\in\mathbb Z$, which says $x\sim x$.

Let $x\sim y$ and $y\sim z$.

Thus, $x-y\in\mathbb Z$ and $y-z\in\mathbb Z$, which gives $$x-z=x-y+y-z\in\mathbb Z,$$ which says $x\sim z$ and we are done!